who invented the amusement park cardboard CARICATURES that you stick
your head through for a photograph

This is a tough question!

I haven't even been able to find out what these things are called -
which makes searching pretty difficult.

Someone else has been seeking a name for them too:
http://www.photo-seminars.com/bbs/messages/49.html

The suggestions he got were "front drop" and "photo popper" which
didn't help, though "photo popper" led me to these pictures of some
(three quarters of the way down the page, four for $550)
http://www.photo-seminars.com/Classified/classads/ads2.htm

The website for the National Amusement Park Historical Association
(NAPHA) is full of interesting information but no answer to your
question, I'm afraid:
http://www.napha.org/faq.html

I agree this one is going to be tough if not impossible. 

"fairground cutouts" or "celebrity cutouts" as they are informally
referred to are also named "photo ops". But they actually predate both
cameras and patenting in their earliest form.

The reason is that they were originally used in medieval times to
place under stocks that criminals were placed in, carved by carpenters
and painted by artists, they would be used to further ridicule the
condemned.

Of course I cannot prove any of this as it is a compilation of scraps
from books, rather than information gathered online, which is thin on
the ground.

That is why I have printed it as a comment to possibly help other
researchers that may have more luck now they have a name to give to
the subject.

Unfortunately, whomever the original inventor or inventors were, I'm
afraidd I struck out.

This question is being asked by a local Radio Station and I have found
all other answers to previous questions by looking under "Radio
Trivia" etc. such as this site
http://www.b101radio.com/morning/iquestlist.asp
Perhaps this will help a researcher?  I increased the answer to be
worth $20 for a tad bit more incentive?